If x 1, what is the value of (-|x| - 1)/(x - 1) ?

The data is sufficient if we get a unique value for the expression \(\frac{-|x| - 1}{x - 1}\)
A quick info about this expression. If x is negative then the expression will have a unique value, which is 1.
If x is 0, it will have a unique value, which is 1.
If x is positive it will have multiple values. For instance, if x = 2, the value of the expression is -3. If x = 3, the value of the expression -2.

So, at some level if we can establish that x is not positive, we will have a unique value. Else we will not have a unique value.

Statement 1: \(\sqrt{x^6} > x^3\)
\(\sqrt{x^6}\) is non negative for real x and will be |x^3|.
For instance \(\sqrt{2^6}\) = 8 and \(\sqrt{{(-2)^6}\) will also be 8.
But 2^3 will be 8 and (-2)^3 will be -8.
So, if we know that \(\sqrt{x^6} > x^3\), we can infer that x is negative.

If x is negative, the value of \(\frac{-|x| - 1}{x - 1}\) will be 1. Take any negative x and check it. Negative integer, negative non integer. It will work for all values.

So, statement 1 ALONE is sufficient.

Statement 2:\(\frac{|x|}{16} + \frac{x}{8} + \frac{|x|}{4} + \frac{x}{2} = -1\)
We can rewrite the expression as |x| + 2x + 4|x| + 8x = -16
5|x| + 10x = -16.
The sum of the expression is -16, which is negative.
5|x| cannot be negative. So, 10x has to be negative if the sum is -16 => x has to be negative.

If x is negative, the expression \(\frac{-|x| - 1}{x - 1}\) has a unique value which is 1.

Statement 2 ALONE is sufficient.

Each statement is INDEPENDENTLY sufficient.
Choice D.